A Statistical Problem Arising in the Theory of Detection of Signals in the Presence of Noise in a Multi-Channel System and Leading to Stable Distribution Laws

The following statistical problem arising in the theory of radar is solved. There are two hypotheses concerning joint densities of random variables ξ ₁ , ⋯ ξ ₙ\[ {gathered} A:P_A (x_1 , ⋯ ,x_n ) = ∏i = 1^n {{{2x_i }}{d}e^{ - x_i^{{2 / d}} } } , \\ B:P_B (x_1 , ⋯ ,x_n ) = 1/n∑j = 1^n {{{2x_j }}{{d + d}}e^{ - x_d^{{2 {d + d}}} } } ∏i ≠ j {{{2x_i }}{d}e^{ - x_i^{{2 / d}} } } , \\ {gathered} \] where $d > 0$, d > 0. Let β = d / d > 0. Denote by β ₙ (F,D) such a number that there are no tests for distinguishing between hypotheses A and B with error probabilities which are smaller than F and D for β β ₙ (F,D) and that tests exist for β > β ₙ (F,D). It is proved that for n = const, D = const and F → 0\[ β _n (F,D) = {{log n + log {1 / F}}} {{log {1 / D}}} - 1 + o(1). \]An asymptotic formula containing stable distributions for β ₙ (F,D), in which F = const, D = const and n → ∞, is derived.